We now check \(\delta \)  is nonzero.
Locally around a point of \(C\) , \(\delta \) 
can be identified with the derivative of the period map
[1]}
sending a curve corresponding to a fiber of
\(f|_X : X \rightarrow C\) 
to the corresponding Hodge structure on its first cohomology. To show this derivative is not identically zero it suffices to show that the period map is non-constant.
More concretely, by the Torelli theorem, we only need to check
\(f|_X: X \rightarrow C\)  is not isotrivial.
This follows by de Franchis' theorem, as explained in the proof of
theorem:counterexample.

